Move segments of an sphere apart from each other
Select all of your objects in object mode and choose Object -> Transform -> Origin to Geometry.

Then the origins will move to the center of each piece.

Then in the bottom header, to the right of the Object Mode drop down, is a button with a double-headed arrow beneath three dots. Turning this one will "Move Object Origins only..." Then, when you scale it will only scale their POSITION, not their SIZE.

Don't forget to turn it back off.